We say that the operator T on a Hilbert space H into itself is strongly stable if \left\Vert T^nx\right\Vert →0 as n→∞, for allx∈H . If T is a contraction, then T is said to be cs-stable if T has C_0 completely non-unitary part. This note considers the strong stability of operators A⊗B and the p-hyponormality of operators A⊗B. It is shown that the contraction A⊗B is cs-stable if and only if so are the contractionscA and c^{−1}B for some scalar c andA⊗B is p-hyponormal if and only if A andB are. We also characterize p-hyponormal A⊗B for which the commutator |A⊗B|_{2p}−|A^*⊗B^*|^{2p} is compact.